Integral Geometry Methods on Deformed Categories in Field Theory II

Francisco Bulnes

doi:doi:10.11648/j.pamj.s.2014030602.11

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Integral Geometry Methods on Deformed Categories in Field Theory II

Francisco Bulnes

Published in Pure and Applied Mathematics Journal (Volume 3, Issue 6-2)

Received: 8 October 2014 Accepted: 11 October 2014 Published: 24 October 2014

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Abstract

The integral geometry methods are applied on deformed categories to obtain correspondences in the geometrical Langlands program and construct the due equivalences between geometrical objects of the moduli stacks and algebraic objects of the corresponding categories and their L_(G-opers) characterizing the solution classes to field theory equations in the belonging cohomological context such as H^0 (g[[z] ],V_critical )=C[Op_LG (D^X)] which is natural in the framework of the integral transforms to the generalizing of the Zuckerman functors that will be useful to the obtaining of the different factors of the universal functor of derived sheaves of Harish-Chandra to the Langlands geometrical program in mirror symmetry. The cosmological problem that exists is to reduce the number of field equations that are resoluble under the same gauge field (Verma modules) and to extend the gauge solutions to other fields using the topological groups symmetries that define their interactions. This extension can be given by a global Langlands correspondence between the Hecke sheaves category H_(G^^ ) ∞ on an adequate moduli stack and the holomorphic L_(G-) bundles category with a special connection (Deligne connection). The corresponding 〖D 〗_-modules may be viewed as sheaves of conformal blocks (or co-invariants) (images under a generalized version of the Penrose transform) naturally arising in the framework of conformal field theory.

Published in	Pure and Applied Mathematics Journal (Volume 3, Issue 6-2) This article belongs to the Special Issue Integral Geometry Methods on Derived Categories in the Geometrical Langlands Program
DOI	10.11648/j.pamj.s.2014030602.11
Page(s)	1-5
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Langlands Correspondence, Hecke Sheaves Category, Moduli Stacks, Ramifications, Twisted 〖D 〗_- Modules, Verma Module Extensions

References

[1]	F. Bulnes, Cohomology of Moduli Spaces in Differential Operators Classification to the Field Theory (II), in: Proceedings of Function Spaces, Differential Operators and Non-linear Analysis., 2011, Tabarz Thur, Germany, Vol. 1 (12) pp001-022.
[2]	F. Bulnes, Geometrical Langlands Ramifications and Differential Operators Classification by Coherent D-Modules in Field Theory, Journal of Mathematics and System Sciences, David Publishing, USA Vol. 3, no.10, pp491-507.
[3]	F. Bulnes, Penrose Transform on Induced DG/H-Modules and Their Moduli Stacks in the Field Theory, Advances in Pure Mathematics 3 (2) (2013) 246-253. doi: 10.4236/apm.2013.32035.
[4]	A. Kapustin, M. Kreuser and K. G. Schlesinger, Homological mirror symmetry: New Developments and Perspectives, Springer. Berlin, Heidelberg, 2009.
[5]	F. Bulnes, Integral geometry and complex integral operators cohomology in field theory on space-time, in: Proceedings of 1st International Congress of Applied Mathematics-UPVT (Mexico)., 2009, vol. 1, Government of State of Mexico, pp. 42-51.
[6]	F. Bulnes, Penrose Transform on D-Modules, Moduli Spaces and Field Theory, Advances in Pure Mathematics 2 (6) (2012) 379-390. doi: 10.4236/apm.2012.26057.
[7]	F. Bulnes, Cohomology of Moduli Spaces on Coherent Sheaves to Conformal Class of the Space-Time, Technical report for XLIII-National Congress of Mathematics of SMM, (RESEARCH) Tuxtla Gutiérrez, Chiapas, Mexico, 2010.
[8]	M. Kashiwara and W. Schmid, “Quasi-equivariant D-modules, equivariant derived category, and representations of reductive Lie groups, in Lie Theory and Geometry,” Progr. Math. vol. 123, Birkhäuser, Boston, 1994, 457–488.
[9]	M. Kashiwara, On the maximally over determined systems of linear differential equation I*. Publ. R.I.M.S. Kyoto Univ. 10 (1975) 563-579.
[10]	D. Ben-zvi and D. Nadler, The character theory of complex group, 5 (2011) arXiv:0904.1247v2[math.RT],
[11]	D. Ben-Zvi, J. Francis, and D. Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry, J. Amer. Math. Soc. 23 (2010) 909-966. doi:arXiv:0805.0157.
[12]	E. Frenkel, Lectures on the Langlands Program and conformal field theory, Preprint hep-th/0512172.
[13]	A. D’Agnolo and P. Schapira. Radon-Penrose transform for D-modules, J. Funct. Anal. 139 (2) (1996) 349–382.
[14]	A. Grothendieck, On the De Rham Cohomology of algebraic varieties, Publ. Math.I.H.E.S. 29 (1966) 95-103.
[15]	E. Frenkel, Ramifications of the Geometric Langlands Program, CIME Summer School “Representation Theory and Complex Analysis”, Venice, June 2004.
[16]	R. Hartshorne, Deformation Theory (Graduate Texts in Mathematics), Springer, USA, 2010.
[17]	F. Bulnes (2014) Derived Categories in Langlands Geometrical Ramifications: Approaching by Penrose Transforms. Advances in Pure Mathematics, 4, 253-260. doi: 10.4236/apm.2014.46034.
[18]	R. Donagi, T. Pantev, Lectures on the Geometrical Langlands Conjecture and non-Abelian Hodge Theory, 07/01/2008-06/30/2009, , Shing-Tung Yau "Surveys in Differential Geometry 2009", International Press, 2009.
[19]	R. Donagi, T. Pantev (2012), Langlands duality for Hitchin systems, Invent.math. arXiv:math/0604617
[20]	F. Oort, Yoneda extensions in abelian categories, Mathematische Annalen. 01-1964; 153(3):227-235. DOI: 10.1007/BF01360318
[21]	E. Frenkel, C. Teleman, Geometric Langlands correspondence near opers, J. of the Ramanujan Math. Soc. pp. 123–147.
[22]	E. Frenkel, C. Teleman, Self extensions of Verma modules and differential forms on opers, Comps. Math. 142 (2006), 477-500.

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Francisco Bulnes. (2014). Integral Geometry Methods on Deformed Categories in Field Theory II. Pure and Applied Mathematics Journal, 3(6-2), 1-5. https://doi.org/10.11648/j.pamj.s.2014030602.11

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ACS Style

Francisco Bulnes. Integral Geometry Methods on Deformed Categories in Field Theory II. Pure Appl. Math. J. 2014, 3(6-2), 1-5. doi: 10.11648/j.pamj.s.2014030602.11

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AMA Style

Francisco Bulnes. Integral Geometry Methods on Deformed Categories in Field Theory II. Pure Appl Math J. 2014;3(6-2):1-5. doi: 10.11648/j.pamj.s.2014030602.11

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@article{10.11648/j.pamj.s.2014030602.11,
  author = {Francisco Bulnes},
  title = {Integral Geometry Methods on Deformed Categories in Field Theory II},
  journal = {Pure and Applied Mathematics Journal},
  volume = {3},
  number = {6-2},
  pages = {1-5},
  doi = {10.11648/j.pamj.s.2014030602.11},
  url = {https://doi.org/10.11648/j.pamj.s.2014030602.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.s.2014030602.11},
  abstract = {The integral geometry methods are applied on deformed categories to obtain correspondences in the geometrical Langlands program and construct the due equivalences between geometrical objects of the moduli stacks and algebraic objects of the corresponding categories and their L_(G-opers) characterizing the solution classes to field theory equations in the belonging cohomological context such as H^0 (g[[z] ],V_critical )=C[Op_LG (D^X)] which is natural in the framework of the integral transforms to the generalizing of the Zuckerman functors that will be useful to the obtaining of the different factors of the universal functor of derived sheaves of Harish-Chandra to the Langlands geometrical program in mirror symmetry. The cosmological problem that exists is to reduce the number of field equations that are resoluble under the same gauge field (Verma modules) and to extend the gauge solutions to other fields using the topological groups symmetries that define their interactions. This extension can be given by a global Langlands correspondence between the Hecke sheaves category H_(G^^ ) ∞ on an adequate moduli stack and the holomorphic L_(G-) bundles category with a special connection (Deligne connection). The corresponding 〖D 〗_-modules may be viewed as sheaves of conformal blocks (or co-invariants) (images under a generalized version of the Penrose transform) naturally arising in the framework of conformal field theory.},
 year = {2014}
}

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TY  - JOUR
T1  - Integral Geometry Methods on Deformed Categories in Field Theory II
AU  - Francisco Bulnes
Y1  - 2014/10/24
PY  - 2014
N1  - https://doi.org/10.11648/j.pamj.s.2014030602.11
DO  - 10.11648/j.pamj.s.2014030602.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 1
EP  - 5
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.s.2014030602.11
AB  - The integral geometry methods are applied on deformed categories to obtain correspondences in the geometrical Langlands program and construct the due equivalences between geometrical objects of the moduli stacks and algebraic objects of the corresponding categories and their L_(G-opers) characterizing the solution classes to field theory equations in the belonging cohomological context such as H^0 (g[[z] ],V_critical )=C[Op_LG (D^X)] which is natural in the framework of the integral transforms to the generalizing of the Zuckerman functors that will be useful to the obtaining of the different factors of the universal functor of derived sheaves of Harish-Chandra to the Langlands geometrical program in mirror symmetry. The cosmological problem that exists is to reduce the number of field equations that are resoluble under the same gauge field (Verma modules) and to extend the gauge solutions to other fields using the topological groups symmetries that define their interactions. This extension can be given by a global Langlands correspondence between the Hecke sheaves category H_(G^^ ) ∞ on an adequate moduli stack and the holomorphic L_(G-) bundles category with a special connection (Deligne connection). The corresponding 〖D 〗_-modules may be viewed as sheaves of conformal blocks (or co-invariants) (images under a generalized version of the Penrose transform) naturally arising in the framework of conformal field theory.
VL  - 3
IS  - 6-2
ER  -

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Author Information

Francisco Bulnes

Head of Research Department, Research Department in Mathematics and Engineering, TESCHA, Federal Highway, Mexico-Cuautla Tlapala “La Candelaria”, Chalco, P. C. 56641, Mexico

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APA Style

Francisco Bulnes. (2014). Integral Geometry Methods on Deformed Categories in Field Theory II. Pure and Applied Mathematics Journal, 3(6-2), 1-5. https://doi.org/10.11648/j.pamj.s.2014030602.11

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ACS Style

Francisco Bulnes. Integral Geometry Methods on Deformed Categories in Field Theory II. Pure Appl. Math. J. 2014, 3(6-2), 1-5. doi: 10.11648/j.pamj.s.2014030602.11

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AMA Style

Francisco Bulnes. Integral Geometry Methods on Deformed Categories in Field Theory II. Pure Appl Math J. 2014;3(6-2):1-5. doi: 10.11648/j.pamj.s.2014030602.11

Copy | Download

@article{10.11648/j.pamj.s.2014030602.11,
  author = {Francisco Bulnes},
  title = {Integral Geometry Methods on Deformed Categories in Field Theory II},
  journal = {Pure and Applied Mathematics Journal},
  volume = {3},
  number = {6-2},
  pages = {1-5},
  doi = {10.11648/j.pamj.s.2014030602.11},
  url = {https://doi.org/10.11648/j.pamj.s.2014030602.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.s.2014030602.11},
  abstract = {The integral geometry methods are applied on deformed categories to obtain correspondences in the geometrical Langlands program and construct the due equivalences between geometrical objects of the moduli stacks and algebraic objects of the corresponding categories and their L_(G-opers) characterizing the solution classes to field theory equations in the belonging cohomological context such as H^0 (g[[z] ],V_critical )=C[Op_LG (D^X)] which is natural in the framework of the integral transforms to the generalizing of the Zuckerman functors that will be useful to the obtaining of the different factors of the universal functor of derived sheaves of Harish-Chandra to the Langlands geometrical program in mirror symmetry. The cosmological problem that exists is to reduce the number of field equations that are resoluble under the same gauge field (Verma modules) and to extend the gauge solutions to other fields using the topological groups symmetries that define their interactions. This extension can be given by a global Langlands correspondence between the Hecke sheaves category H_(G^^ ) ∞ on an adequate moduli stack and the holomorphic L_(G-) bundles category with a special connection (Deligne connection). The corresponding 〖D 〗_-modules may be viewed as sheaves of conformal blocks (or co-invariants) (images under a generalized version of the Penrose transform) naturally arising in the framework of conformal field theory.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - Integral Geometry Methods on Deformed Categories in Field Theory II
AU  - Francisco Bulnes
Y1  - 2014/10/24
PY  - 2014
N1  - https://doi.org/10.11648/j.pamj.s.2014030602.11
DO  - 10.11648/j.pamj.s.2014030602.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 1
EP  - 5
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.s.2014030602.11
AB  - The integral geometry methods are applied on deformed categories to obtain correspondences in the geometrical Langlands program and construct the due equivalences between geometrical objects of the moduli stacks and algebraic objects of the corresponding categories and their L_(G-opers) characterizing the solution classes to field theory equations in the belonging cohomological context such as H^0 (g[[z] ],V_critical )=C[Op_LG (D^X)] which is natural in the framework of the integral transforms to the generalizing of the Zuckerman functors that will be useful to the obtaining of the different factors of the universal functor of derived sheaves of Harish-Chandra to the Langlands geometrical program in mirror symmetry. The cosmological problem that exists is to reduce the number of field equations that are resoluble under the same gauge field (Verma modules) and to extend the gauge solutions to other fields using the topological groups symmetries that define their interactions. This extension can be given by a global Langlands correspondence between the Hecke sheaves category H_(G^^ ) ∞ on an adequate moduli stack and the holomorphic L_(G-) bundles category with a special connection (Deligne connection). The corresponding 〖D 〗_-modules may be viewed as sheaves of conformal blocks (or co-invariants) (images under a generalized version of the Penrose transform) naturally arising in the framework of conformal field theory.
VL  - 3
IS  - 6-2
ER  -

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